Importance of Graph Evaluation in Shortest Path Algorithms

CIS121-204 – Fall 2007 Lab 12 – Last Lab Tuesday, December 4, 2007

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Graph • Graph G=(V,E) consists of a set of vertices V and a set of edges E, where E is a subset of VxV. • vertex: singular vertices: plural • G is simple if it contains no self loops or parallel edges. • Multigraph is graph with parallel edges.

Graph: Example 1 • Graph 1 is a multigraph with a self loop at vertex 4. • Vertex 0 is adjacent to vertices 1 and 2. • Graph 1 is not connected; it has two connected components: {0,1,2,3} and {4}. • Edges of Graph 1 are weighted because each edge has a “weight.” • Graph 1 is undirected because edges have no directions. 2550 0 1 100 2547 14 11 2007 3 4 2 Graph 1 V={0,1,2,3,4} E={(0,1),(0,2),(0,2),(1,3), (2,3),(4,4)} Note that G1 is a multigraph, so E is a multiset (having duplicate elements).

Graph: Example 2 • Graph 2 is a directed graph because edges have directions. • Graph 2 is simple because there is at most one edge in the same direction between any two vertices. • Edges of Graph 2 are unweighted—all edges have the same “weight,” say 1. • Graph 2 is connected because for any two vertices u and v, there is a path from u to vor from v to u. • If for any two vertices u and v, there is a path from u to vand from v to u, the graph is strongly connected. 0 1 2 3 4 5 6 Graph 2

Adjacency Lists • If there is an edge from u to v, then v is in the adjacency list of u. • What’s the adjacency lists of this graph? • The first few: 0: {2} 1: {0,4} … 0 1 2 3 4 5 6

Single-Source Shortest Paths • Minimize the “cost” (or length) to go from a “source” vertex to any other vertex. • The length of shortest path from a vertex to itself is zero. • If there is no path from u to v, then the length of shortest path from u to v is infinity. • Weighted graph: CIS320—Dijkstra’s Algorithm • Unweighted graph: Now.

Single-Source Shortest Pathon Unweighted, Directed Graphs • Input: G and source vertex u • Starting from u, try to expand the set reachable vertices. • Have a queue of discovered but unprocessed vertices. • For each element in the queue: • If a new vertex w is discovered, then know the shortest path from u to w. Put w in the queue • If an old vertex is encountered, do nothing. • Let’s do an example.

SSSP on Unweighted Graph: Example 0 ∞ • Want to find the length of shortest paths from source vertex 0. • Assign the initial lengths: 0 for vertex 0 and infinity otherwise • Enqueue 0 because we already “discovered” vertex 0 but has not processed it. • That’s all for the first step. 0 1 ∞ ∞ ∞ 2 3 4 5 6 ∞ ∞ Queue: [0]

SSSP on Unweighted Graph: Example (cont.) 0 ∞ • Dequeue: 0 • Length: 0 • Consider the neighbor of 0 • 2: undiscovered • Update length. • Enqueue 2. 0 1 ∞ 1 ∞ 2 3 4 5 6 ∞ ∞ Queue: [2]

SSSP on Unweighted Graph: Example (cont.) 0 ∞ • Dequeue: 2 • Length: 1 • Consider the neighbor of 2 • 0: discovered • Do nothing. • 3: undiscovered • Update length. • Enqueue 3. • 5: undiscovered • Update length. • Enqueue 5. 0 1 2 1 ∞ 2 3 4 5 6 2 ∞ Queue: [3,5]

SSSP on Unweighted Graph: Example (cont.) 0 3 • Dequeue: 3 • Length: 2 • Consider the neighbor of 3 • 1: undiscovered • Update length. • Enqueue 1. • 4: undiscovered • Update length. • Enqueue 4. 0 1 2 1 3 2 3 4 5 6 2 ∞ Queue: [5,1,4]

SSSP on Unweighted Graph: Example (cont.) 0 3 • Dequeue: 5 • Length: 2 • Consider the neighbor of 5 • 6: undiscovered • Update length. • Enqueue 6. 0 1 2 1 3 2 3 4 5 6 2 3 Queue: [1,4,6]

SSSP on Unweighted Graph: Example (cont.) 0 3 • Dequeue: 1 • Length: 3 • Consider the neighbor of 1 • 0: discovered • Do nothing. • 4: discovered • Do nothing. 0 1 2 1 3 2 3 4 5 6 2 3 Queue: [4,6]

SSSP on Unweighted Graph: Example (cont.) 0 3 • Dequeue: 4 • Length: 3 • Consider the neighbor of 4 • None. • Done. 0 1 2 1 3 2 3 4 5 6 2 3 Queue: [6]

SSSP on Unweighted Graph: Example (cont.) 0 3 • Dequeue: 6 • Length: 3 • Consider the neighbor of 6 • 3: discovered • Do nothing. • 4: discovered • Do nothing. • 5: discovered • Do nothing. 0 1 2 1 3 2 3 4 5 6 2 3 Queue: []

SSSP on Unweighted Graph: Example (cont.) 0 3 • Queue is empty. • Done Done. 0 1 2 1 3 2 3 4 5 6 2 3 Queue: []

SSSP on Unweighted Graph: Algorithm • Array sp of length |V| initialized as infinity • sp[u]=0 //the shortest path from u to u has length 0 • Queue Q • Q.enqueue(u) • while Q is not empty • s=Q.dequeue() • for each v such that (s,v) is an edge • if sp[s]+1<sp[v] • sp[v]=sp[s]+1 • Q.enqueue(v) • return sp

Takeaways from CIS121 • Using a correct data structure saves time. • Three steps in writing a program: • Designing • Implementing • Testing You should spend most of your time in the first and last steps. • Eclipse… please. • Real programmers don’t have a life???

What to do after CIS121? • You probably want to see my handwriting on the whiteboard for the last time in this course… • … because I will use a chalkboard for the review session.

Time’s Up. • Thanks for the good time we have had. • Hope you enjoy the section. I always do. • See you in the review session and the final exam. • Reminder: Final Exam • Date: Thursday, December 13, 2007 • Time: 9-11AM • Place: Skirkanich Hall Auditorium • Good luck. • See you around. • Feel free to (and please) say hi.

Importance of Graph Evaluation in Shortest Path Algorithms